18950
J. Phys. Chem. 1996, 100, 18950-18961
Dynamics of Quasi Two-Dimensional Colloidal Systems Jeremy Schofield, Andrew H. Marcus, and Stuart A. Rice* Department of Chemistry and The James Franck Institute, The UniVersity of Chicago, Chicago, Illinois 60635 ReceiVed: April 23, 1996; In Final Form: August 7, 1996X
In this paper we examine the asymptotic long time dynamics of quasi two-dimensional colloidal suspensions over a wide range of concentrations. At low concentrations the dynamics is determined by uncorrelated binary collisions among the constituent particles. These collisions among the particles lead to logarithmic corrections to the well-known linear growth in time of the mean squared displacement of the particles in the suspension. The self-scattering function of the suspension can be related to the mean squared displacement via the Gaussian approximation, which we examine in detail for systems of low concentration. At higher concentrations caging effects influence the dynamics of the suspension, which we account for by developing a formal mode coupling theory for colloidal systems from first principles. Equations for the dynamics of the memory functions that account for caging effects are derived and solved self-consistenly, for the case of instanteous hydrodynamic interactions, by utilizing the Gaussian approximation for the scattering functions of the colloidal system and assuming a particular form for the cumulants of the position. We find that the functional form suggested by Cichocki and Felderhof for the time dependence of the mean squared displacement of quasi two-dimensional colloidal systems in the limit that hydrodynamic interactions are instantaneous is compatible with the predictions of mode coupling theory. Futhermore, we explicitly evaluate the long time diffusion coefficient and other parameters as a function of concentration.
1. Introduction Recent studies of quasi two-dimensional colloidal suspensions using light microscopy have provided detailed information on the dynamics of the suspension over a wide range of concentration.1-3 These data provide information about the microscopic state of the system as a function of time and therefore permit direct calculation of statistical properties such as the mean squared displacement of the colloid particles. It is well-known that the mean squared displacement of an isolated Brownian particle in a three-dimensional fluid increases linearly with time. However a suspension of interacting Brownian particles demonstrates much more complicated behavior,4 in which the mean squared displacements at short and long times grow linearly in time, but the mean squared displacement at intermediate time grows in a complicated fashion that depends on the volume fraction of the suspended particles and on the nature of their interactions. Recently, much attention has been focused on quasi twodimensional (thin cell) colloidal suspensions where the restricted geometry of the system is reflected in both static5 and dynamic properties.2,6 The influence of geometric restrictions on the dynamics of the suspension is an issue of much interest and debate in the scientific community. Many theorists have used the Smoluchowski operator, which is proportional to a Brownian particle transport coefficient, to derive a generalized Langevin equation that describes the dynamics of a system constrained to have a restricted geometry without considering whether the relevant transport coefficient is well-defined. It is well established that in a strictly two-dimensional system consisting of a large Brownian particle immersed in a bath of small discs the mean squared displacement, in fact, grows more strongly than linearly at long times due to mode coupling effects which describe the formation of vortex motions in the fluid.7 The mode coupling effects imply that the motion of the Brownian disc in a two-dimensional system cannot be described by a X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01171-9 CCC: $12.00
diffusion equation and that the hydrodynamic equations do not describe the dynamics of the bath of discs since the transport coefficients do not exist. In apparent contradiction with these results, several authors have succesfully utilized the Smoluchowski equation, which incorporates a diffusion coefficient, in their studies of the dynamics of quasi two-dimensional suspensions and completely ignored both mode coupling effects of the bath and correlated collisions among the Brownian particles. In particular, Cichocki and Felderhof6 recently investigated the self-diffusion in a semidilute suspension of interacting Brownian particles using the adjoint Smoluchowski operator to generate time displacements in the long time regime. They found that the mean squared displacement W(t) behaves for long times t as
W(t) ) DsLt + (DsS - DsL)τL log(t/τm) + o(1)
(1.1)
where DsL and DsS are the long and short time self-diffusion constants and τL and τm are time scales that depend on the nature of the interactions between Brownian particles and the volume fraction of the suspension. They claim that the logarithmic term in (1.1) is due to a 1/t singularity in a memory function that is characteristic of Brownian systems in two dimensions, and hence (1.1) should apply for two-dimensional systems at any concentration. Experimentally this claim appears to have some validity, as is shown in Figure 1. Although these data are for relatively short times compared to the time required for an isolated Brownian particle to diffuse a distance of several diameters at high concentrations, the functional form is readily apparent. A description of these experiments will be published elsewhere.2 In this paper we carefully examine these issues in detail, and we develop a formal description of the dynamics of a colloidal suspension in a thin cell. Our theory develops a proper framework that incorporates the effects of both mode coupling and binary uncorrelated collisions of the Brownian particles on the memory function describing the Brownian dynamics in a simple model system of hard spheres. We demonstrate that the functional form of the dynamics of the suspension is determined © 1996 American Chemical Society
Dynamics of Quasi Two-Dimensional Colloidal Systems
J. Phys. Chem., Vol. 100, No. 49, 1996 18951 Hb is the Hamiltonian for the isolated bath molecules, and φ describes the interactions between the bath and colloid particles. The Hamiltonian HB has the form
HB(X) )
PN‚PN + U(RN) 2M
(2.2)
where M is the mass of the colloid particles and U can be written as a sum of two-body short-ranged potentials, N
U(RN) ) ∑ ∑U(Rij) Figure 1. Plot of the mean squared displacement W(t) divided by time t versus log(t)/t at reduced areal densities F* ) nσ2/As, where σ is the diameter of the spheres, n is the number density, and As is the area of the system. The letters in the diagram correspond to the reduced areal density values A ) 0.08, B ) 0.24, C ) 0.50, D ) 0.58, E ) 0.69, F ) 0.84, and G ) 0.88.
principally by the binary collisions, although the model coupling has significant effects on the long time self-diffusion constant of the system. The paper is organized as follows: In section 2 the model system is introduced and a systematic method of calculating correlation functions involving only Brownian particle degrees of freedom is presented. We derive the Fokker-Planck operator for time displacements of the Brownian degrees of freedom at times greater than the time scale for decay of the bath particle correlations. In section 3 we present the mode coupling formalism; it is based upon N ordering techniques developed elsewhere.8 We formulate an expression for the generalized collective and self-diffusion coefficients and derive selfconsistent equations for the memory functions that appear in the expressions for the generalized diffusion coefficients. We then examine these expressions in the asymptotic long time diffusive regime where correlations involving the momenta have decayed, and we calculate the self-diffusion coefficient. At low concentrations the mode coupling effects are small and the selfdiffusion coefficient is determined mainly by binary uncorrelated collisions. At higher concentrations, where the mode coupling effects are significant, we investigate the compatibility of a functional form proposed by Cichocki and Felderhof6 (eq 1.1) for the time dependence of the mean squared displacement with mode coupling theory. We find that although the mode coupling effects significantly decrease the magnitude of the long time self-diffusion coefficient, the effects are established rapidly and to a good approximation do not alter the functional dependence on the time of the mean squared displacement. In section 4 we suumarize these results.
where Rij ≡ |Ri - Rj|. The Hamiltonian Hb is of the form N*
Hb ) ∑ j)1
(
πj‚πj 2m
+ V(rj)
)
(2.4)
The Liouville operator for the system can be written as
L ) LB + Lb + Lφ
(2.5)
where
LB ) -
PN ∇ + ∇RNU∇PN M RN
Lb ) -
πN ∇ + ∇rN*V∇πN* m rN*
(2.6)
Lφ ) ∇RNφ∇PN + ∇rN*φ∇πN*
(2.7)
and
The Liouville operator L determines the time evolution of an arbitrary function of the variables of the system, C(X,Xb), according to
C(X,Xb;t) ) e-LtC(X,Xb)
(2.8)
We are interested in the long time behavior of correlation functions of dynamical variables which are defined in the colloid phase space X, such as
F(k,t) )
〈N ˆ (k,t) N ˆ (k)*〉F
)
ˆ (k)]N ˆ (k)*〉F 〈[e-LtN
NS(k)
NS(k)
(2.9)
where S(k) is the structure factor for the colloid particles,
2. Fokker-Planck Dynamics The quasi two-dimensional colloidal system consists of N identical massive spherical particles of radius a whose center of mass coordinates and momenta are denoted by RN and PN, respectively, where RN and PN are 2N-dimensional vectors with components Rj and Pj with j ) 1, ..., N. The colloid particles are immersed in a simple fluid composed of N* small molecules and the entire system is confined between two plates. The geometry of the system restricts the center of mass motion of the colloid particles to a plane, but allows the solvent molecules to move in three dimensions. We denote the phase point for the colloid degrees of freedom by X ≡ (RN, PN) and the phase point for the solvent degrees of freedom by Xb ≡ (rN*, πN*). The Hamiltonian for the system can be written in the form
H(X,Xb) ) HB(X) + Hb(Xb) + φ(X,Xb)
(2.3)
i)1 j>i
(2.1)
where HB is the Hamiltonian for the isolated colloid particles,
N
N ˆ (k) ) (1 - δk,0) ∑ e-ik‚Rj
(2.10)
j)1
and 〈B〉F denotes the average of B(X) over the equilibrium distribution function of the system
Fe )
e-βHe-βHbe-βφ q
(2.12)
where
q ≡ ∫ dX dXb e-βHe-βHbe-βφ In eq 2.12 we have used the conventional notation β ≡ (KBT)-1, with T being the temperature of the overall system. We define
18952 J. Phys. Chem., Vol. 100, No. 49, 1996
Schofield et al.
the reduced distribution function for the colloid subsystem We by integrating Fe over the bath phase space,
We(RN,PN) ) ∫dXb Fe(X,Xb)
(2.12)
and define F˜(RN,Xb) to be the conditional distribution function for the bath molecules in the presence of fixed colloid particles;
F˜ ≡ Fe/We
(2.13)
Note that F˜ is normalized in the sence that ∫dXb F˜ ) 1. Since Fe ) F˜We, we may write for a correlation function involving dynamical variables of the X phase space
〈[e-Lt A(X)]B(X)〉F ) 〈[〈e-Lt〉bA(X)]B(X)〉
(2.14)
where 〈f(X,Xb)〉b ) ∫dXb F˜(RN,Xb) f(X,Xb) and 〈B(X)〉 ) ∫dX B(X) We(X). Using projection operator techniques,8 it is straightforward to establish that
∫0∞dt e-zt〈e-Lt〉b ) ∫0∞dt e-ztΞ(t) ) Ξ˜ (z) ) z - O1 † (z)
(2.15)
where
PN ‚∇RN + M β ∇pi - Pi ‚Γ ˜ ij(RN,z)‚∇pj (2.16) M
O† (z) ) -∇RN(U + ω)‚∇PN +
∑ i,j
(
)
In eq 2.16, ω(RN) is the potential of mean force,
e-βω(R ) ) ∫dXb Fb(Xb)e-βφ N
(2.17)
where Fb is the equilibrium distribution function for the isolated bath molecules. The generalized friction coefficient Γ ˜ ij(RN,z) is defined by the correlation function
ˆ Riφ] ∇ ˆ Rjφ〉b Γ ˜ ij(RN,z) ) ∫0 dt e-zt〈[e-QbLt∇ ∞
(2.18)
ˆ riφ ) ∇Ri(φ - ω), and PbB(X) ) where Qb ) (1 - Pb), ∇ 〈B(X)〉b. Defining the small parameter 2 ) m/M, the generalized friction coefficient can be approximated as8
ˆ Riφ]∇ Γ ˜ ij(RN,z) ) ∫0 dt e-zt 〈[e-L0t∇ ˆ Rjφ〉b + O() ∞
(2.19)
where L0 ) Lb + ∇rN*φ∇πN*. If bath molecule correlations decay on a time scale τb that is short relative to the time scale of the decay of colloid particle correlations, then for t . τb †
Ξ(t) ≈ eO t
(2.20)
where O† is the Fokker-Planck operator,
PN ‚∇RN + M β ˜ ij(RN,z)0)‚∇Pj (2.21) ∇pi - Pi ‚Γ M
O† ) - ∇RN (U + ω)‚∇PN +
∑ i,j
(
)
These results suggest that when t . τb the time evolution of colloid particle correlation functions is determined by the Fokker-Planck operator O† according to †
〈A(X;t) B(X)〉F ) 〈[eO t A(X)]B(X)〉
(2.22)
where A(X) and B(X) are arbitrary dynamical variables of the colloid particle phase space. It should be noted that in a strictly two-dimensional system of colloid discs and solvent discs the friction coefficient Γ ˜ ij(RN,z) does not exist as z approaches zero due to mode coupling effects; in that case no separation of time scales exists between bath particle and colloid particle correlation function decays. For finite and three-dimensional systems the coupling of bath particle modes to colloid particle motion yields finite contributions and allows the z ) 0 friction coefficient to be related to properties of the solvent, such as its viscosity. These “hydrodynamic” interactions also lead to a complicated functional dependence of Γ ˜ ij(RN,z) on the spatial positions RN of the colloid particles;9 they are known to be important in determining the transport properties of colloid systems.10 Throughout this paper we will assume that the hydrodynamic interactions occur on a time scale much shorter than the time scale of interactions among the colloid particles and hence will be treated as instantaneous. Medina-Noyola21 has suggested a simple picture of the dynamics of suspensions in which between collisions a colloid particle diffuses in a static field of its neighbors. The presence of other particles modifies only the effective friction a particle feels while diffusing, hence the identification
(KBT)2
〈Γ ˜ ij〉 ) δi,jI
Dss(φ)
(2.23)
is made, where Dss(φ) is the density dependent “short time” self-diffusion coefficient. The short time diffusion coefficient determines the dynamics of the self-scattering Fs(k,t) for times shorter than τI, the average time a colloid particle requires to diffuse a mean separation distance between colloid particles in ˜, the suspension. Typically, one expects τI ≈ 1/nD0 . MKbT/Γ where D0 is the diffusion coefficient for an isolated colloid particle in solution. Ideally, Dss(φ) can be calculated given a particular model, but in practice it is simpler to determine Dss(φ) empirically from fitted data. In the following sections ˜ ij ≡ Iδi,jD unspecified, keeping in mind that we leave (KbT)2/Γ D should be closely related to Dss(φ). 3. Projection Operator Formalism and Mode Coupling In this section we develop a systematic mode coupling formalism that in principle allows the transport properties and dynamics of colloid suspensions of arbitrary concentration to be calculated. Our view of the dynamics of colloid suspensions borrows heavily from knowledge of the character of the long time dynamics of collective excitations in hard sphere liquids. Over the last 15 years considerable progress has been made in the study of hard sphere liquids. de Schepper, Cohen, and coworkers have established the existence of short-wavelength collective excitations in hard sphere liquids that determine the dynamic scattering function F(k,t) at intermediate times (i.e. on the order of 10 collision times).11 They have shown that these collective excitations, which are extensions to nonzero wavevector of the hydrodynamic modes of the fluid, give important mode coupling contributions and are essential for the proper description of the dynamics of density fluctuations at intermediate times. The important physical parameters that characterize the collective excitations are the density n and the mean-free-path l. The collective excitations lose their significance when kl g 1. At high concentrations we generally expect l . a, where a is the radius of the hard sphere; hence collective excitations are important in concentrated colloid systems even for ka ≈ 1.
Dynamics of Quasi Two-Dimensional Colloidal Systems
J. Phys. Chem., Vol. 100, No. 49, 1996 18953
Generally it is assumed that the long time behavior of density fluctuations is determined by a set of slow modes of the system. Initially it was thought that the only slow modes of a liquid system were the densities of the conserved quantities of the system in the small wavevector regime. The long time behavior of the system can then be represented in terms of the dynamics of those important “hydrodynamic” densities.13 It is now apparent that in dense systems collective modes extended to large wavevectors, up to Kc ≈ 1/l, are still slower than the other “kinetic” modes of the system, which implies that the cutoff wavevector Kc is larger than initially thought to be the case. This realization helps explain why the magnitude of the long time tail of a correlation function calculated in a molecular dynamics simulation is, typically, much larger than the first mode coupling calculations predicted.12 In analogy with hard sphere liquid systems, we shall assume that the long time dynamics of the density fluctuations in colloid suspensions is determined by the extensions of the conserved hydrodynamic densities to larger “nonhydrodynamic” wavevectors. For the colloid suspension these variables are the number N e-ik‚Rj and the momentum density N ˆ (k) ) (1 - δk,0) ∑j)1 N -ik‚Rj . We define our complete set of density P(k) ) ∑j)1Pje slow variables to be A(k) ) {N ˆ (k),P(k)} and
†
f(k,t) ) O†e(1-P1)O t(1 - P1)O†A(k)
Noting that Q1(k) ) A(k), we find for the Fourier-Laplace transform of the scattering function F(k,t)
F ˜ (k,z) )
Qi(k) ) A(k)
D ˜ (k,z) )
KbT
1 MS(k) z + (1/mKbT) (γ1(k) + η˜ (k,z)/n)
η˜ (k,z) )
Q1(k) - 〈A(k-q) A(q)〉δk,0
where K11(k) ) 〈Q1(k) Q1(k)*〉. The subtractions are included in (3.1) to ensure that the basis set is orthogonal in mode order in the sense that KRβ ) 〈QRQ* β〉 ) δ|R|,|β|KRβ. We define the projection operators P1 and P by -1
P1B(k) ) 〈B(k) Q1(k)*〉K11(k)
[
O†P(k) ) ∑ j)1
(3.2)
where the * notation denotes a sum over mode orders, over repeated wavevectors, and over hydrodynamic (i.e. N and P) indices. We now apply the operator identity
e(A+B)t ) eAt + ∫0 dτ eA(t-τ) Be(A+B)τ t
(3.3)
to the expressions exp (1 - P1)t and exp(1 - P)t, and we obtain13
〈Q(z)Q*〉 ) ∫0 dt e-zt〈Q(t)Q*〉 ) [zI - M(z)]-1*K ∞
〈A(k,z) A(k)*〉 ) [zI - M ˜ (k,z)]-1K11(k) ) [zI - M(z)]-1 11 K11(k) (3.4) where -1
- 〈φ˜ (z)Q*〉*K
M ˜ (k,z) ) 〈[O† A(k)]A(k)*〉K11(k)-1 - 〈f˜(k,z) A(k)*〉K11(k)-1 (3.6) and
φ(t) ) O†e(1-P)O t(1 - P)O†Q
Pi‚Γ ˜ ije-ik‚R ∑ i,j
(3.7)
) K(k) ) -ik‚τ(k) - γ2(k)
we find
η˜ (k,z) ) -
1
Tr 〈Γ ˜ ije-ik‚(R -R )〉 ∑ 2N i,j i
j
(3.13)
† ∞ 1 dt e-zt 〈[e(1-P1)O tKD1 (k)]‚KD1 (k)*〉 d ∫0 2L
where KD1 (k) ) (1 - P1)K(k). When the hydrodynamic ˜ , and γ1(k) ) Γ ˜, interactions are instantaneous Γ ˜ ij ) IδijΓ KD1 (k) ) -ik‚(1 - P1)τ(k). Since τ(k) is the stress tensor for the colloid particle suspension, we see that η˜ (k,z) is proportional to k2 times the Green-Kubo expression for the viscosity of the suspension. Mode coupling equations can be systematically derived by applying N ordering techniques to the generalized hydrodynamic matrix M(z) in eq 3.5, noting that the multilinear matrix MRβ(z) can be factorized for the colloid subsystem in the same way as for a simple liquid or granular system. The nonlinear Fokker-Planck time displacement operator does not alter the analysis since it can be shown that terms such as
˜ ij‚∇R A(q) ∇R A(k-q)‚Γ ∑ i,j i
(3.5)
(3.12)
]
j
M
ik ‚P(k) M
ik ‚PjPj + Fj e-ik‚Rj M
γ1(k) )
Q1(k)
PB(k) ) 〈B(k)Q*)*K-1*Q
†
(3.11)
-1 〈f˜(k,z)‚P(k)*〉 2Ld
ˆ (k) ) O† N
β
M(z) ) 〈[O Q]Q*〉*K
1 〈[O†P(k)]‚P(k)*〉 2N
γ1(k) )
N
-1
(3.10)
and
l
†
(3.9)
with a generalized collective “diffusion” coefficient
(3.1)
Q2(k,q) ) A(k-q) A(q) 〈A(k-q) A(q) Q1(k)*〉K11(k)
1 z + k2D ˜ (k,z)
Using the facts that
Q0 ) 1
-1
(3.8)
j
(3.14)
are negligible in N order compared to terms such as (O†A(kq))A(q). Details of the N ordering formalism can be found in ref 13. Using the N ordering technique, a self-consistent expression for the transport coefficient η˜ (k,z) can be obtained by expanding the multilinear correlation functions in terms of the parameter M/N ≈ (Kcψ)2, where ψ is the correlation length of the two-dimensional system. The series is formally
18954 J. Phys. Chem., Vol. 100, No. 49, 1996
Schofield et al.
resummed to yield a self-consistent expression for η˜ (k,z). The mode coupling series can be written as
η˜ (k,z) ) η˜ B(k,z) + η˜ MC(k,z)
simple liquid system.13 For the suspension, the static vertices in eqs 3.19 and 3.20 are given by
(3.15)
E M ˆ P;NN (k,q)
)
〈(O†P(k))QNN(k-q,q)*〉 N2S(k-q) S(q)
where E (k,q) ) M ˆ NN;P
† ∞ 1 η˜ (k,z) ) - d ∫0 dt e-zt〈[e(1-P)O tKD(k)]‚KD(k)*〉 (3.16) 2L
B
has all the projections onto the set of slow variables Q removed from its fluctuating force KD(k) ) (1 - P)K(k). Note that, in contrast, KD1 (k) has only the linear projections of the slow variables removed from K(k). The mode coupling term may be represented as an infinite series of terms written either in self-consistent form or in an expansion in powers of M/N. In self-consistent form this series looks like
〈(O†NN(k-q,q))P(k)*〉 NMKbT
The analysis just elaborated for the scattering function F(k,t) can be extended to tagged particle correlations like the Fourier transform of the van-Hove self-correlation function
ˆ 1(k,t) N ˆ 1(k)*〉 ) 〈e-ik‚(R1(t)-R1(0))〉 (3.22) Fs(k,t) ) 〈N Applying the projection operator and mode coupling formalism to the Laplace transform Fs(k,t) gives
∞
(i) (k,z) η˜ (k,z) ) ∑ΘPP
1 z+k D ˜ s(k,z)
(3.23)
KbT 1 M z + (β/M)(〈Γ ˜ 11〉 + η˜ s(k,z))
(3.24)
F ˜ s(k,z) )
(3.17)
i)2
D ˜ s(k,z) )
where ∞
(2) (k,z) ) ΘPP
∑ XPR *GR |R |)2 1
1
(z)‚XR1P
(3.18)
η˜ s(k,z) ) - ∫0 dt ∞
∞
∑
|R∞1|,|R2)2
2
where
1
(3) ΘPP (k,z) )
XPR1*GR1(z)‚XR1R2*GR2(z)‚XR2P KsD((k) ) (1 - P1)
e-zt (1-P1)O†t D+ 〈[e Ks (k)]‚KsD-(k)*〉 d 2L
‚P P ( F ]e [-ik M
-ik‚R1
1 1
1
- (1 - P1)γs2(k)
l where the vertex XRβ is defined to be MRβ(z) with at least one wavevector inequality between the sets R and β. The propagators GR(z) are the Laplace transforms of products of the full hydrodynamic correlation functions such as F(k,t), 〈P(k,t) P(k)*〉, and 〈P(k,t) N ˆ (k)*〉. The vertices XRβ contain static parts that are expressible in terms of static correlation functions and “dissipative” parts which resemble transport coefficients. The simplest mode coupling terms involves only the bilinear modes and is given by
η˜ MC(k,z) ) MKbT E E M ˆ P;NN (k,q) M ˆ NN;P (k,q) Ld (3.19) dq ∫ d 2 B (2π) z + (k - q) D ˜ (k-q,z) + q2D ˜ B(q,z)
where D ˜ B(k,z) is defined by (3.10) with η˜ (k,z) ) η˜ B(k,z). The mode coupling series can also be formally resummed to obtain self-consistent equations for the transport coefficients. For example, the simplest bilinear contribution in self-consistent form is
ηMC(k,t) ) MKbT Ld E E (k,q) M ˆ NN;P (k,q) F(k-q,t) F(q,t) (3.20) ∫dq Mˆ P;NN (2π)d This bilinear term is the most important in determining the nonanalytic k and z dependence of η˜ (k,q) in generalized hydrodynamics, as it gives the leading k and z contributions and is responsible for the t-d/2 long time tail in a d-dimensional
(3.21)
(3.25) with
γs2(k) )
β
N
∑ Pi‚Γ˜ i1e-ik‚R
1
M i)1
(3.26)
Again when the hydrodynamic interactions are instantaneous, ˜ I, (1 - P1)γs2(k) ) 0, and 〈Γ ˜ 11〉 ) Γ ˜ , and the tagged Γ ˜ ij ) δijΓ particle transport coefficient η˜ s(k,z) has a term of order k0, unlike the N particle transport coefficient η˜ (k,z), which is at least of order k2. This difference is due to the fact that the instantaneous force on an individual particles is nonzero even though the total instanteous force on the system is always zero for a conservative system. Mode coupling equations can be derived for the tagged particle transport terms in a manner analogous to that presented above.14 We find that the simplest bilinear mode coupling term in self-consistent form is
ηsMC(k,t) ) MKbT Ld dq M ˆ PsE1;N1N(k,q) M ˆ NsE1N;P1(k,q) Fs(k-q,t) F(q,t) (3.27) d∫ (2π) where
M ˆ PsE1;N1N(k,q) M ˆ NsE1N;P1(k,q)
)
〈[O†P1(k)]QN1N(k-q,q)*〉 NS(q)
)
〈[O†QN1N(k-q,q)]P1(k)*〉 MKbT
(3.28)
(3.29)
Dynamics of Quasi Two-Dimensional Colloidal Systems Note that, through (3.27), the dynamics of Fs(k,t) is related to the dynamics of the N-particle correlation function F(k,t). When the hydrodynamic interactions are instanteous, we find the products of the tagged particle and N-particle vertices are given by
M ˆ PsE1;N1N(k,q)
M ˆ NsE1N;P1(k,q)
)
KbT(nqh(q))2 MN(1 + nh(q))
(3.30)
E E (k,q) M ˆ NN;P (k,q) ) M ˆ P;NN
KbTn2 S(|k-q|) S(q) ((k - q)C(|k-q|) + qC(q))2 (3.31) MN where in eq 3.31 we have approximated the three-point correlation function with the relation
〈N ˆ (k) N ˆ (k-q)* N ˆ (q)*〉 ≈ NS(|k-q|) S(q) S(k)
(3.32)
In eq 3.30, N is the total number of colloid particles in the system, n ) N/Ld is the number density of colloid particles, nh(q) ) S(q) - 1, and C(q) ) 1/n - 1/(nS(q)) is the direct correlation function of the colloid system. Note that as q grows large, the vertex functions approach zero and hence naturally cutoff the integral over the wavevector at large q. It should be noted that eq 3.20 for ηMC(k,t) has been derived systematically and is identical in form to the mode coupling equations derived by Szamel and Lowen in their discussion of the glass transition in colloid systems.18 The mode coupling terms are designed to take into consideration the fact that at intermediate to high densities each particle is surrounded by a cage of others, which leads to correlated collisions and collective vortex motions in the suspension. The B “bare” term η˜ (s) (k,z) has all the correlated motion removed D from its fluctuating force K(s) (k,t) via the projection operator (1 - P) and hence can be thought of as having its dynamics determined by uncorrelated binary collisions alone. For lowdensity systems the upper cutoff Kc is small since the mean distance over which a particle diffuses before colliding with another particle is large, and hence one expects the mode coupling contributions to be negligible. As the density increases the cutoff increases and the vertex functions contain more significant structure, which in turn implies that the mode coupling terms become important. Near the close packed density one anticipates that the feedback mechanism can lead to anamolous increases in the memory term of the generalized friction coefficient and can provide a mechanism for critical slowing down and glass formation.
4. Hard Sphere Dynamics in the Diffusive Regime 4.1. Low-Density Systems. To solve the hierarchy of equations for the correlation functions Fs(k,t) and F(k,t), some B approximate form of the bare memory functions η˜ (s) (k,z) must be adopted. The simplest sensible approximation for the bare memory function is to use the low-density form for all densities. We shall concern ourselves mainly with correlation functions in the “diffusive” time regime t . τD, where τD ) M/βΓ ˜ , and assume that the hydrodynamic interactions are instantaneous. This time scale is the so-called “overdamped” limit in which
J. Phys. Chem., Vol. 100, No. 49, 1996 18955 terms of order zτD are ignored,15,16 and it corresponds to setting
D ˜ s(k,z) )
D ˜ (k,z) )
(KbT)2 Γ ˜ + η˜ s(k,z)
)D
Γ ˜ Γ ˜ + η˜ s(k,z)
(KbT)2
1 Γ ˜ D ) S(k) Γ ˜ + η˜ (k,z)/n S(k) Γ ˜ + η˜ (k,z)/n
(4.1)
˜. in eqs 3.10 and 3.23, where we have defined D ) (KbT)2/Γ Furthermore, we shall assume that correlation functions involving momenta of the particles decay more rapidly than do correlation functions that involve only spatial coordinates. Under these conditions we find that, by averaging over the momenta of the system,
η˜ B(k,z)/n ) η˜ sB(k,z) )
k2 ∞ -zt QNΩ†t uD dt e 〈[e τl (k)]‚τluD(k)*〉 d ∫0 2L † ∞ 1 dt e-zt 〈[eQNΩ tFD1 (k)]‚FD1 (k)*〉 (4.2) d ∫0 2L
where QN ) (1 - PN) removes the multilinear products of the density,
τluD(k) ) QN
1
∑ kˆ ‚Fij(e-ik‚R - e-ik‚R ) i
j
2 i,j*i
FD1 (k) ) QN ∑ kˆ ‚F1je-ik‚Ri
(4.3)
j*1
Fij ) - ∇Ri (U(Rij) + ω(Rij)), and Ω† is the one-particle irreducible Smoluchowski operator given by17,18 N
Ω† ) D ∑ (∇Ri + βFi)‚Qi∇Ri
(4.4)
i)1
In eq 4.4, the projection operator Qi is defined by QiB(k) ) B(k) - 〈B(k)e-ik‚Ri〉e-ik‚Ri. The one-particle irreducible operator is obtained from17 †
†
〈eQO t〉P ) eQNΩ t
(4.5)
where 〈f(P)〉P ) ∫dPN f(P)Πm(PN) is the average of f(P) over the Maxwell distribution Πm(PN). We shall assume that the propagator exp(QNΩ†t) represents the dynamics of uncorrelated binary collisions among the particles and replace QNΩ† by Ω†2, where
Ω†2 ) D(∇R12 + ∇R22 + βF12‚(∇R1 - ∇R2))
(4.6)
is the two-particle Smoluchowski operator. Separating the twoparticle Smoluchowski equation into an equation for the center of mass coordinate R ) (R1 + R2)/2 and the relative coordinate r ) R1 - R2, we find that, for a hard sphere interaction, to leading order in the area fraction φ ) nπa2 of the spheres (of radius a) in the suspension, the two-dimensional transport
18956 J. Phys. Chem., Vol. 100, No. 49, 1996
Schofield et al.
η(k,t)/n ) g(2a) -(ka)2t/2τ ∞ 0 4φΓ ˜ e du e-ut/τ0 [2J′0(ka)2 P0(ka,u) + ∫ 0 τ0
coefficients η˜ B(k,z)/n and η˜ sB(k,z) are given by
η˜ sB(k,z) ) 4φg(2a)
(KbT)2 2
4a
∞
∫dr dr0 δ(r0-2a) δ(r-2a) A˜ (k,r-r0,z) ×
4 ∑ J′2l(ka)2 P2l(ka,u)] l)1
eik/2‚(r-r0) (kˆ ‚rˆ )(kˆ ‚rˆ 0) η˜ B(k,z)/n ) (KbT)2 4φg(2a) ∫dr dr0 δ(r0-2a) δ(r-2a) A˜ (k,r-r0,z) × 4a2 (kˆ ‚rˆ )(kˆ ‚r0)(e-ik/2‚r0 - eik/2‚r0)(eik/2‚r - e-ik/2‚r) (4.7) where g(2a) is the pair correlation function for the suspension evaluated at the contact distance r ) 2a and
∫0 dt e-zt∫d(R-R0) eik‚(R-R ) A(R-R0,r-r0,t) ∞
0
and Jl(x) and Nl(x) are Bessel functions. Thus for low-density systems, in which mode coupling contributions are negligible, we find that the long time self-diffusion constant is
[
∞
)
(KbT)2 Γ ˜ (1 + 2φg(2a))
I′m(2ay˜ )
]
m
Fs(k,t) ) 〈e-ik‚(R1(t)-R1(0))〉 ) 1 exp{-k2W(t)} 1 + R2(t)(k2W(t))2 + O(k6) (4.16) 2
[
(4.10)
where ) max{r,r0}, ) min{r,r0}, θ and θ0 are the polar angles of r and r0 in the plane, respectively, y˜ ) (z/2D + k2/4)1/2, and Im and Km are modified Bessel functions. With this form of A ˜ (k,r-r0,z), we finally obtain ∞
Kl(y)
∑ J′l(ka)2 -yK′l(y) l)-∞
∞
˜ g(2a) η˜ B(k,z)/n ) 4φΓ
R2(t) )
Kl(y) l
where y ) 2ay˜ ) (2zτ0 + (ka)2)1/2 with τ0 ) a2/D. In the limit k ) z ) 0, these bare memory functions are
˜ 2φg(2a) η˜ sB(0,0) ) Γ
(4.12)
and η˜ B(0,0) ) 0, whereas η˜ B(k,z)/k2 diverges logarithmically as k and z approach zero. In the time domain the bare memory functions are given by
ηsB(k,t) ) g(2a) -(ka)2t/2τ ∞ 0 e 4φΓ ˜ ∫0 du e-ut/τ0 [J′0(ka)2 P0(ka,u) + τ0 ∞
2 ∑ J′l(ka)2 Pl(ka,u)] (4.13) l)1
〈(∆R1(t))4〉 2〈(∆R1(t))2〉2
-1
(4.17)
with ∆R1(t) ) R1(t) - R1(0). Note that for two-dimensional systems in which the displacements ∆R1(t) have a Gaussian distribution
(4.11)
(1 + (-1)l)J′l(ka)2 ∑ -yK′(y) l)-∞
]
where W(t) ) 〈(R1(t) - R1(0))2〉/4 is the mean squared displacement and R2(t) is defined by
r
) Im(y˜ r
